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Statistic identifier: St000184
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Collection: Integer partitions
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Description: The size of the centralizer of any permutation of given cycle type.
The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$:
$$C_g = \{h \in G : hgh^{-1} = g\}.$$
Its size thus depends only on the conjugacy class of $g$.
The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is
$$|C| = \Pi j^{a_j} a_j!$$
For example, for any permutation with cycle type $\lambda = (3,2,2,1)$,
$$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$
There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.
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References:
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Code:
def statistic(p):
return p.centralizer_size()
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Statistic values:
[] => 1
[1] => 1
[2] => 2
[1,1] => 2
[3] => 3
[2,1] => 2
[1,1,1] => 6
[4] => 4
[3,1] => 3
[2,2] => 8
[2,1,1] => 4
[1,1,1,1] => 24
[5] => 5
[4,1] => 4
[3,2] => 6
[3,1,1] => 6
[2,2,1] => 8
[2,1,1,1] => 12
[1,1,1,1,1] => 120
[6] => 6
[5,1] => 5
[4,2] => 8
[4,1,1] => 8
[3,3] => 18
[3,2,1] => 6
[3,1,1,1] => 18
[2,2,2] => 48
[2,2,1,1] => 16
[2,1,1,1,1] => 48
[1,1,1,1,1,1] => 720
[7] => 7
[6,1] => 6
[5,2] => 10
[5,1,1] => 10
[4,3] => 12
[4,2,1] => 8
[4,1,1,1] => 24
[3,3,1] => 18
[3,2,2] => 24
[3,2,1,1] => 12
[3,1,1,1,1] => 72
[2,2,2,1] => 48
[2,2,1,1,1] => 48
[2,1,1,1,1,1] => 240
[1,1,1,1,1,1,1] => 5040
[8] => 8
[7,1] => 7
[6,2] => 12
[6,1,1] => 12
[5,3] => 15
[5,2,1] => 10
[5,1,1,1] => 30
[4,4] => 32
[4,3,1] => 12
[4,2,2] => 32
[4,2,1,1] => 16
[4,1,1,1,1] => 96
[3,3,2] => 36
[3,3,1,1] => 36
[3,2,2,1] => 24
[3,2,1,1,1] => 36
[3,1,1,1,1,1] => 360
[2,2,2,2] => 384
[2,2,2,1,1] => 96
[2,2,1,1,1,1] => 192
[2,1,1,1,1,1,1] => 1440
[1,1,1,1,1,1,1,1] => 40320
[9] => 9
[8,1] => 8
[7,2] => 14
[7,1,1] => 14
[6,3] => 18
[6,2,1] => 12
[6,1,1,1] => 36
[5,4] => 20
[5,3,1] => 15
[5,2,2] => 40
[5,2,1,1] => 20
[5,1,1,1,1] => 120
[4,4,1] => 32
[4,3,2] => 24
[4,3,1,1] => 24
[4,2,2,1] => 32
[4,2,1,1,1] => 48
[4,1,1,1,1,1] => 480
[3,3,3] => 162
[3,3,2,1] => 36
[3,3,1,1,1] => 108
[3,2,2,2] => 144
[3,2,2,1,1] => 48
[3,2,1,1,1,1] => 144
[3,1,1,1,1,1,1] => 2160
[2,2,2,2,1] => 384
[2,2,2,1,1,1] => 288
[2,2,1,1,1,1,1] => 960
[2,1,1,1,1,1,1,1] => 10080
[1,1,1,1,1,1,1,1,1] => 362880
[10] => 10
[9,1] => 9
[8,2] => 16
[8,1,1] => 16
[7,3] => 21
[7,2,1] => 14
[7,1,1,1] => 42
[6,4] => 24
[6,3,1] => 18
[6,2,2] => 48
[6,2,1,1] => 24
[6,1,1,1,1] => 144
[5,5] => 50
[5,4,1] => 20
[5,3,2] => 30
[5,3,1,1] => 30
[5,2,2,1] => 40
[5,2,1,1,1] => 60
[5,1,1,1,1,1] => 600
[4,4,2] => 64
[4,4,1,1] => 64
[4,3,3] => 72
[4,3,2,1] => 24
[4,3,1,1,1] => 72
[4,2,2,2] => 192
[4,2,2,1,1] => 64
[4,2,1,1,1,1] => 192
[4,1,1,1,1,1,1] => 2880
[3,3,3,1] => 162
[3,3,2,2] => 144
[3,3,2,1,1] => 72
[3,3,1,1,1,1] => 432
[3,2,2,2,1] => 144
[3,2,2,1,1,1] => 144
[3,2,1,1,1,1,1] => 720
[3,1,1,1,1,1,1,1] => 15120
[2,2,2,2,2] => 3840
[2,2,2,2,1,1] => 768
[2,2,2,1,1,1,1] => 1152
[2,2,1,1,1,1,1,1] => 5760
[2,1,1,1,1,1,1,1,1] => 80640
[1,1,1,1,1,1,1,1,1,1] => 3628800
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Created: May 04, 2014 at 23:41 by Lahiru Kariyawasam
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Last Updated: Oct 29, 2017 at 16:33 by Martin Rubey